Question 1

Use Taylor's formula to find the requested approximation of f(x, y) near the origin.
-Cubic approximation to $f ( x , y ) = e ^ { x + 2 y }$

Accepted Answer

A)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 2 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } y + \frac { 4 } { 3 } x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
B)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 1 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + 1 x ^ { 2 } y + 2 x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
C)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 1 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } y + \frac { 4 } { 3 } x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
D)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 2 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + 1 x ^ { 2 } y + 2 x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
A)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 2 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } y + \frac { 4 } { 3 } x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
B)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 1 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + 1 x ^ { 2 } y + 2 x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
C)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 1 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } y + \frac { 4 } { 3 } x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$ 
D)  $1 + x + 2 y + \frac { 1 } { 2 } x ^ { 2 } + 2 x y + 2 y ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + 1 x ^ { 2 } y + 2 x y ^ { 2 } + \frac { 4 } { 3 } y ^ { 3 }$

Question 2

Find the extreme values of the function subject to the given constraint.
-$f ( x , y ) = x y , \quad 9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Accepted Answer

A)  Maximum: 3 at $\left( \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
B)  Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
C)  Maximum: 3 at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
D)  Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
A)  Maximum: 3 at $\left( \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
B)  Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
C)  Maximum: 3 at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ 
D)  Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$

Question 3

Solve the problem.
-Find the derivative of the function $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 }$ at the point $( 4,5 )$ in the direction in which the function increases most rapidly.

Accepted Answer

A)  $\sqrt { 293 }$ 
B)  $3 \sqrt { 42 }$ 
C)  $\sqrt { 299 }$ 
D)  $\sqrt { 365 }$ 
A)  $\sqrt { 293 }$ 
B)  $3 \sqrt { 42 }$ 
C)  $\sqrt { 299 }$ 
D)  $\sqrt { 365 }$

Question 4

Solve the problem.
-Find the least squares line through the points (1, -4) and (2, 2).

Accepted Answer

A)  y = 6x - 10 
B)  y = -2x - 10 
C)  y = -2x - 6 
D)  y = 6x - 6 
A)  y = 6x - 10 
B)  y = -2x - 10 
C)  y = -2x - 6 
D)  y = 6x - 6

Question 5

Find the extreme values of the function subject to the given constraint.
-$$f ( x , y ) = x y , \quad x ^ { 2 } + y ^ { 2 } = 32$$

Accepted Answer

A)  Maximum: 16 at (4, 4); minimum: -16 at (-4, -4) 
B)  Maximum: 16 at (4, -4) and (-4, 4); minimum: -16 at (4, 4) and (-4, -4) 
C)  Maximum: 16 at (4, 4) and (-4, -4); minimum: -16 at (4, -4) and (-4, 4) 
D)  Maximum: 16 at (4, 4); minimum: 0 at (0, 0) 
A)  Maximum: 16 at (4, 4); minimum: -16 at (-4, -4) 
B)  Maximum: 16 at (4, -4) and (-4, 4); minimum: -16 at (4, 4) and (-4, -4) 
C)  Maximum: 16 at (4, 4) and (-4, -4); minimum: -16 at (4, -4) and (-4, 4) 
D)  Maximum: 16 at (4, 4); minimum: 0 at (0, 0)

Question 6

Estimate the error in the quadratic approximation of the given function at the origin over the given region.
-$$f ( x , y ) = \sin 3 x \sin ^ { 2 } 4 y , | x | \leq 0.1 , | y | \leq 0.1$$

Accepted Answer

A)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.018$ 
B)  $| E ( x , y ) | \leq 0.064$ 
C)  $| E ( x , y ) | \leq 0.256$ 
D)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.0853$ 
A)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.018$ 
B)  $| E ( x , y ) | \leq 0.064$ 
C)  $| E ( x , y ) | \leq 0.256$ 
D)  $| \mathrm { E } ( \mathrm { x } , \mathrm { y } ) | \leq 0.0853$

Question 7

Solve the problem.
-About how much will $f ( x , y , z ) = x y ^ { 3 } z ^ { 2 }$ change if the point $( x , y , z )$ moves from $( 3 , - 7 , - 6 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction of $2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ ?

Accepted Answer

A)  $\frac { 2646 } { 5 }$ 
B)  $\frac { 882 } { 5 }$ 
C)  $- \frac { 2646 } { 5 }$ 
D)  $- \frac { 882 } { 5 }$ 
A)  $\frac { 2646 } { 5 }$ 
B)  $\frac { 882 } { 5 }$ 
C)  $- \frac { 2646 } { 5 }$ 
D)  $- \frac { 882 } { 5 }$

Question 8

Give an appropriate answer.
-$f ( x , y ) = x ^ { 3 } - 7 x ^ { 2 } y - 6 x y ^ { 3 }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 18 x y ^ { 2 }$ 
B)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } + 2 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } + 3 x y ^ { 2 }$ 
C)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } - 14 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 18 x y ^ { 2 }$ 
D)  $\frac { \partial f } { \partial x } = x ^ { 2 } - 7 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 6 x y ^ { 2 }$ 
A)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 18 x y ^ { 2 }$ 
B)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } + 2 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } + 3 x y ^ { 2 }$ 
C)  $\frac { \partial f } { \partial x } = 3 x ^ { 2 } - 14 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 18 x y ^ { 2 }$ 
D)  $\frac { \partial f } { \partial x } = x ^ { 2 } - 7 x y - 6 y ^ { 3 } ; \frac { \partial f } { \partial y } = - 7 x ^ { 2 } - 6 x y ^ { 2 }$

Question 9

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). -We say that a function $f ( x , y , z )$ approaches the limit $L$ as $( x , y , z )$ approaches $( x 0 , y 0 , z 0 )$ and write $( x , y , z ) \rightarrow \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right) \quad f ( x , y , z ) = L$ if for every number $\varepsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $( x , y , z )$ in the domain of $f$, $0 < \sqrt { \left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } + \left( z - z _ { 0 } \right) ^ { 2 } } < \delta \Rightarrow | f ( x , y , z ) - L | < \varepsilon$. Show that the $\delta - \varepsilon$ requirement in this definition is equivalent to $0 < \left| x - x _ { 0 } \right| < \delta , 0 < \left| y - y _ { 0 } \right| < \delta$, and $0 < \left| \mathrm { z } - \mathrm { z } _ { 0 } \right| < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) - \mathrm { L } | < \varepsilon$.

Accepted Answer

The sphere centered at @#LAT-DLM& of radius @#LAT-DLM& circu

Question 10

At what points is the given function continuous?
-$f ( x , y ) = \tan ( x + y )$

Accepted Answer

A)  All $( x , y ) \neq \left( \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ 
B)  All $( x , y ) \neq ( 0,0 )$ 
C)  All $( x , y )$ such that $x + y \neq \frac { ( 2 n + 1 ) \pi } { 2 }$, where $n$ is an integer 
D)  All ( $x , y )$ 
A)  All $( x , y ) \neq \left( \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ 
B)  All $( x , y ) \neq ( 0,0 )$ 
C)  All $( x , y )$ such that $x + y \neq \frac { ( 2 n + 1 ) \pi } { 2 }$, where $n$ is an integer 
D)  All ( $x , y )$

Question 11

Determine whether the given function satisfies a Laplace equation.
-$$f ( x , y , z ) = 5 x ^ { 2 } - 3 y ^ { 2 } - 2 z ^ { 2 }$$

Accepted Answer

A) True 
 B)False

Question 12

At what points is the given function continuous?
-$$f ( x , y , z ) = y z \cos \left( \frac { 1 } { x } \right)$$

Accepted Answer

A)  All $( x , y , z )$ such that $0 \leq \frac { 1 } { x } \leq 2 \pi$ 
B)  All $( x , y , z )$ such that $x \neq 0$ 
C)  All $( x , y , z )$ 
D)  All $( x , y , z )$ such that $x \neq \frac { 2 } { \pi }$ 
A)  All $( x , y , z )$ such that $0 \leq \frac { 1 } { x } \leq 2 \pi$ 
B)  All $( x , y , z )$ such that $x \neq 0$ 
C)  All $( x , y , z )$ 
D)  All $( x , y , z )$ such that $x \neq \frac { 2 } { \pi }$

Question 13

Use Taylor's formula to find the requested approximation of f(x, y) near the origin.
-Quadratic approximation to $f ( x , y ) = \frac { 1 } { 1 + 6 x + y }$

Accepted Answer

A)  $1 - 6 x - y + 36 x ^ { 2 } + 6 x y + y ^ { 2 }$ 
B)  $1 - 6 x - y + 36 x ^ { 2 } + 12 x y + y ^ { 2 }$ 
C)  $1 + 6 x + y + 36 x ^ { 2 } + 12 x y + y ^ { 2 }$ 
D)  $1 + 6 x + y + 36 x ^ { 2 } + 6 x y + y ^ { 2 }$ 
A)  $1 - 6 x - y + 36 x ^ { 2 } + 6 x y + y ^ { 2 }$ 
B)  $1 - 6 x - y + 36 x ^ { 2 } + 12 x y + y ^ { 2 }$ 
C)  $1 + 6 x + y + 36 x ^ { 2 } + 12 x y + y ^ { 2 }$ 
D)  $1 + 6 x + y + 36 x ^ { 2 } + 6 x y + y ^ { 2 }$

Question 14

Solve the problem.
-Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (-2, -4, -6) in the direction in which the function decreases most rapidly.

Accepted Answer

A)  $- \frac { 5 } { 34 } \sqrt { 2 }$ 
B)  $- \frac { 5 } { 26 } \sqrt { 2 }$ 
C)  $- \frac { 5 } { 14 } \sqrt { 2 }$ 
D)  $- \frac { 5 } { 22 } \sqrt { 2 }$ 
A)  $- \frac { 5 } { 34 } \sqrt { 2 }$ 
B)  $- \frac { 5 } { 26 } \sqrt { 2 }$ 
C)  $- \frac { 5 } { 14 } \sqrt { 2 }$ 
D)  $- \frac { 5 } { 22 } \sqrt { 2 }$

Question 15

Solve the problem.
-Find an equation for the level surface of the function $f ( x , y , z ) = \ln \left( \frac { x y } { z } \right)$ that passes through the point $\left( \mathrm { e } ^ { 4 } , \mathrm { e } ^ { 12 } , \mathrm { e } ^ { 3 } \right)$

Accepted Answer

A)  $\frac { x y } { z } = e ^ { 16 }$ 
B)  $\ln \left( \frac { x y } { z } \right) = 16$ 
C)  $\frac { x y } { z } = e ^ { 13 }$ 
D)  $\ln \left( \frac { x y } { z } \right) = \frac { 1 } { 16 }$ 
A)  $\frac { x y } { z } = e ^ { 16 }$ 
B)  $\ln \left( \frac { x y } { z } \right) = 16$ 
C)  $\frac { x y } { z } = e ^ { 13 }$ 
D)  $\ln \left( \frac { x y } { z } \right) = \frac { 1 } { 16 }$

Question 16

Find the limit.
-$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 x ^ { 2 } + 4 y ^ { 2 } + 1 } { 10 x ^ { 2 } - 4 y ^ { 2 } + 4 }$$

Accepted Answer

A)  1 
B)  $- 1$ 
C)  $\frac { 1 } { 4 }$ 
D)  No limit 
A)  1 
B)  $- 1$ 
C)  $\frac { 1 } { 4 }$ 
D)  No limit

Question 17

Solve the problem.
-Find the equation for the tangent plane to the surface -7x - 6y - 5z = -11 at the point (1, -1, 2).

Accepted Answer

A)  -7x + 6y - 10z = -18 
B)  -7x + 6y - 10z = -11 
C)  -7x - 6y - 5z = -11 
D)  -7x - 6y - 5z = -18 
A)  -7x + 6y - 10z = -18 
B)  -7x + 6y - 10z = -11 
C)  -7x - 6y - 5z = -11 
D)  -7x - 6y - 5z = -18

Question 18

Solve the problem.
-Find the point on the line 2x + 3y = 5 that is closest to the point (1, 2).

Accepted Answer

A)  $\left( \frac { 5 } { 17 } , \frac { 15 } { 17 } \right)$ 
B)  $\left( \frac { 7 } { 17 } , \frac { 13 } { 17 } \right)$ 
C)  $\left( \frac { 7 } { 13 } , \frac { 17 } { 13 } \right)$ 
D)  $\left( \frac { 5 } { 13 } , \frac { 15 } { 13 } \right)$ 
A)  $\left( \frac { 5 } { 17 } , \frac { 15 } { 17 } \right)$ 
B)  $\left( \frac { 7 } { 17 } , \frac { 13 } { 17 } \right)$ 
C)  $\left( \frac { 7 } { 13 } , \frac { 17 } { 13 } \right)$ 
D)  $\left( \frac { 5 } { 13 } , \frac { 15 } { 13 } \right)$

Question 19

Solve the problem.
-Evaluate $\frac { \mathrm { dw } } { \mathrm { dt } }$ at $\mathrm { t } = 5$ for the function $\mathrm { w } = \mathrm { e } ^ { \mathrm { y } } - \ln \mathrm { x } ; \mathrm { x } = \mathrm { t } ^ { 2 } , \mathrm { y } = \ln \mathrm { t }$.

Accepted Answer

A)  3 
B)  $- \frac { 3 } { 5 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $\frac { 4 } { 5 }$ 
A)  3 
B)  $- \frac { 3 } { 5 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $\frac { 4 } { 5 }$

Question 20

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = \frac { \cos y } { x z ^ { 2 } }$

Accepted Answer

A)  $f _ { x } = - \frac { \cos y } { x ^ { 2 } z ^ { 2 } } ; f _ { y } = - \frac { \sin y } { x z _ { 2 } ^ { 2 } } ; f _ { z } = - \frac { 2 \cos y } { x z ^ { 3 } }$ 
B)  $\mathrm { f } _ { \mathrm { x } } = \frac { \cos \mathrm { y } } { \mathrm { z } ^ { 2 } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \sin \mathrm { y } } { \mathrm { xz } ^ { 2 } } ; \mathrm { f } _ { \mathrm { z } } = \frac { 2 \cos \mathrm { y } } { \mathrm { xz } }$ 
C)  $f _ { x } = - \frac { \cos y } { z ^ { 2 } } ; f _ { y } = - \frac { \sin y } { x z ^ { 2 } } ; f _ { z } = - \frac { 2 \cos y } { x z }$ 
A)  $f _ { x } = - \frac { \cos y } { x ^ { 2 } z ^ { 2 } } ; f _ { y } = - \frac { \sin y } { x z _ { 2 } ^ { 2 } } ; f _ { z } = - \frac { 2 \cos y } { x z ^ { 3 } }$ 
B)  $\mathrm { f } _ { \mathrm { x } } = \frac { \cos \mathrm { y } } { \mathrm { z } ^ { 2 } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \sin \mathrm { y } } { \mathrm { xz } ^ { 2 } } ; \mathrm { f } _ { \mathrm { z } } = \frac { 2 \cos \mathrm { y } } { \mathrm { xz } }$ 
C)  $f _ { x } = - \frac { \cos y } { z ^ { 2 } } ; f _ { y } = - \frac { \sin y } { x z ^ { 2 } } ; f _ { z } = - \frac { 2 \cos y } { x z }$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to $f ( x , y ) = e ^ { x + 2 y }$

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = x y , \quad 9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Solve the problem. -Find the derivative of the function $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 }$ at the point $( 4,5 )$ in the direction in which the function increases most rapidly.

Solve the problem. -Find the least squares line through the points (1, -4) and (2, 2).

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = x y , \quad x ^ { 2 } + y ^ { 2 } = 32$

Estimate the error in the quadratic approximation of the given function at the origin over the given region. - $f ( x , y ) = \sin 3 x \sin ^ { 2 } 4 y , | x | \leq 0.1 , | y | \leq 0.1$

Solve the problem. -About how much will $f ( x , y , z ) = x y ^ { 3 } z ^ { 2 }$ change if the point $( x , y , z )$ moves from $( 3 , - 7 , - 6 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction of $2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ ?

Give an appropriate answer. - $f ( x , y ) = x ^ { 3 } - 7 x ^ { 2 } y - 6 x y ^ { 3 }$

At what points is the given function continuous? - $f ( x , y ) = \tan ( x + y )$

Determine whether the given function satisfies a Laplace equation. - $f ( x , y , z ) = 5 x ^ { 2 } - 3 y ^ { 2 } - 2 z ^ { 2 }$

At what points is the given function continuous? - $f ( x , y , z ) = y z \cos \left( \frac { 1 } { x } \right)$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Quadratic approximation to $f ( x , y ) = \frac { 1 } { 1 + 6 x + y }$

Solve the problem. -Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (-2, -4, -6) in the direction in which the function decreases most rapidly.

Solve the problem. -Find an equation for the level surface of the function $f ( x , y , z ) = \ln \left( \frac { x y } { z } \right)$ that passes through the point $\left( \mathrm { e } ^ { 4 } , \mathrm { e } ^ { 12 } , \mathrm { e } ^ { 3 } \right)$

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 x ^ { 2 } + 4 y ^ { 2 } + 1 } { 10 x ^ { 2 } - 4 y ^ { 2 } + 4 }$

Solve the problem. -Find the equation for the tangent plane to the surface -7x - 6y - 5z = -11 at the point (1, -1, 2).

Solve the problem. -Find the point on the line 2x + 3y = 5 that is closest to the point (1, 2).

Solve the problem. -Evaluate $\frac { \mathrm { dw } } { \mathrm { dt } }$ at $\mathrm { t } = 5$ for the function $\mathrm { w } = \mathrm { e } ^ { \mathrm { y } } - \ln \mathrm { x } ; \mathrm { x } = \mathrm { t } ^ { 2 } , \mathrm { y } = \ln \mathrm { t }$ .

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = \frac { \cos y } { x z ^ { 2 } }$

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Exam 15: Partial Derivatives

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to f(x,y)=ex+2yf ( x , y ) = e ^ { x + 2 y }f(x,y)=ex+2y

Find the extreme values of the function subject to the given constraint. - f(x,y)=xy,9x2+4y2=36f ( x , y ) = x y , \quad 9 x ^ { 2 } + 4 y ^ { 2 } = 36f(x,y)=xy,9x2+4y2=36

Solve the problem. -Find the derivative of the function f(x,y)=x2+xy+y2f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 }f(x,y)=x2+xy+y2 at the point (4,5)( 4,5 )(4,5) in the direction in which the function increases most rapidly.